MA 1111: Linear Algebra I
Homework problems due October 29, 2015
Solutions to this problem sheet are to be handed in after our class at 3pm on Thursday. Please attach a cover sheet with a declarationhttp://tcd-ie.libguides.com/plagiarism/declaration confirming that you know and understand College rules on plagiarism.
Please put your name and student number on each of the sheets you are handing in. (Or, ideally, staple them together).
1.Find all i,j,k,lfor which the permutation
5 2 k 3 l 1 4 1 3 i 6 j
is even.
2. (a) Using the property tr(AB) = tr(BA) (see the previous homework), prove that if A and Bare two n×n-matrices, and Ais invertible, then tr(A−1BA) =tr(B).
(b) Use the previous question to show that if for two n×n-matrices A and B we have AB−BA=A, thenAis not invertible.
3.Compute the determinant of the matrix (a)
1 0 −2 1 1 3 4 3 1
; (b)
1 1 −2 −1 2 0 3 −1 4 2 3 1 3 0 0 1
. 4.For which values of cdoes Afail to be invertible:
(a) A=
2−c −1
−1 2−c
; (b) A=
2 1 3
c−1 2+c 4c
1 3 −1
.
5.Which of the following statements are true (for n×n-matrices):
1. If ABis invertible, then Ais invertible and Bis invertible;
2. If Ais invertible and Bis invertible, then ABis invertible;
3. If AB=0 then eitherA=0 or B=0;
4. If AB=0 then neitherAnorBis invertible;
5. If AB=0 then at least one ofAand Bis not invertible.
(In these statements,0means the matrix whose all entries are equal to zero). For those of these statements which are true, prove them, for the rest give counterexamples.
